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 PolyLog

 http://functions.wolfram.com/10.08.06.0035.01

 Input Form

 PolyLog[n, z] \[Proportional] Zeta[n] + ((z - 1)^(n - 1)/(n - 1)!) (PolyGamma[n] + EulerGamma - ((-1 + EulerGamma (-1 + n) + (-1 + n) PolyGamma[n])/2) (z - 1) - ((-1 + 6 n + (2 + n - 3 n^2) (EulerGamma + PolyGamma[n]))/24) (z - 1)^2 - ((1 - 4 n - 3 n^2 + (-1 + n) (1 + n) (2 + n) (EulerGamma + PolyGamma[n]))/48) (z - 1)^3 + \[Ellipsis]) + Sum[((Zeta[n - j] (z - 1)^j)/j!) (1 - (j/2) (z - 1) + ((j (5 + 3 j))/24) (z - 1)^2 - ((j (2 + j) (3 + j))/48) (z - 1)^3 + \[Ellipsis]), {j, 1, n - 2}] - ((z - 1)^(n - 1)/(n - 1)!) (1 - ((n - 1)/2) (z - 1) + (((-1 + n) (2 + 3 n))/24) (z - 1)^2 - (((-1 + n) (1 + n) (2 + n))/48) (z - 1)^3 + \[Ellipsis]) Log[1 - z] /; (z -> 1) && Element[n, Integers] && n > 3

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Zeta", "[", "n", "]"]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", "n", "]"]], "+", "EulerGamma", "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", RowBox[List["PolyGamma", "[", "n", "]"]]]]]], "2"], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["6", " ", "n"]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n", "-", RowBox[List["3", " ", SuperscriptBox["n", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["PolyGamma", "[", "n", "]"]]]], ")"]]]]]], "24"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "-", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["4", " ", "n"]], "-", RowBox[List["3", " ", SuperscriptBox["n", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["PolyGamma", "[", "n", "]"]]]], ")"]]]]]], "48"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "2"]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["n", "-", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "j"]]]]], RowBox[List["j", "!"]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["j", "2"], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["j", " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["3", " ", "j"]]]], ")"]]]]]], "24"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "-", RowBox[List[FractionBox[RowBox[List["j", " ", RowBox[List["(", RowBox[List["2", "+", "j"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "j"]], ")"]]]], "48"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"]]], " ", "+", "\[Ellipsis]"]], ")"]]]]]], "-", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["3", " ", "n"]]]], ")"]]]], "24"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]], "48"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"]]], " ", "+", "\[Ellipsis]"]], ")"]], RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "3"]]]]]]]]

 MathML Form

 Li PolyLog n ( z ) ζ ( n ) TagBox[RowBox[List["\[Zeta]", "(", TagBox["n", Zeta, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]], Rule[Editable, False], Rule[Selectable, False]] + ( z - 1 ) n - 1 ( n - 1 ) ! ( ψ TagBox["\[Psi]", PolyGamma] ( n ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] - ψ TagBox["\[Psi]", PolyGamma] ( n ) ( n - 1 ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ( n - 1 ) - 1 2 ( z - 1 ) - 6 n + ( - 3 n 2 + n + 2 ) ( ψ TagBox["\[Psi]", PolyGamma] ( n ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) - 1 24 ( z - 1 ) 2 - - 3 n 2 - 4 n + ( n - 1 ) ( n + 1 ) ( n + 2 ) ( ψ TagBox["\[Psi]", PolyGamma] ( n ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) + 1 48 ( z - 1 ) 3 + ) - ( z - 1 ) n - 1 ( n - 1 ) ! ( 1 - n - 1 2 ( z - 1 ) + ( n - 1 ) ( 3 n + 2 ) 24 ( z - 1 ) 2 - ( n - 1 ) ( n + 1 ) ( n + 2 ) 48 ( z - 1 ) 3 + ) log ( 1 - z ) + j = 1 n - 2 ζ ( n - j ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["n", "-", "j"]], Zeta, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]], Rule[Editable, False], Rule[Selectable, False]] ( z - 1 ) j j ! ( 1 - j 2 ( z - 1 ) + j ( 3 j + 5 ) 24 ( z - 1 ) 2 - j ( j + 2 ) ( j + 3 ) 48 ( z - 1 ) 3 + ) /; ( z "\[Rule]" 1 ) n TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n > 3 Condition Proportional PolyLog n z Zeta n z -1 n -1 n -1 -1 PolyGamma n -1 PolyGamma n n -1 n -1 -1 2 -1 z -1 -1 6 n -3 n 2 n 2 PolyGamma n -1 24 -1 z -1 2 -1 -3 n 2 -1 4 n n -1 n 1 n 2 PolyGamma n 1 48 -1 z -1 3 -1 z -1 n -1 n -1 -1 1 -1 n -1 2 -1 z -1 n -1 3 n 2 24 -1 z -1 2 -1 n -1 n 1 n 2 48 -1 z -1 3 1 -1 z j 1 n -2 Zeta n -1 j z -1 j j -1 1 -1 j 2 -1 z -1 j 3 j 5 24 -1 z -1 2 -1 j j 2 j 3 48 -1 z -1 3 Rule z 1 n n 3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Zeta", "[", "n", "]"]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", "n", "]"]], "+", "EulerGamma", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", RowBox[List["PolyGamma", "[", "n", "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox["1", "24"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["6", " ", "n"]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n", "-", RowBox[List["3", " ", SuperscriptBox["n", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["PolyGamma", "[", "n", "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "-", RowBox[List[FractionBox["1", "48"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "n"]], "-", RowBox[List["3", " ", SuperscriptBox["n", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["PolyGamma", "[", "n", "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "2"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["n", "-", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "j"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", "j", " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "24"], " ", RowBox[List["(", RowBox[List["j", " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["3", " ", "j"]]]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "-", RowBox[List[FractionBox["1", "48"], " ", RowBox[List["(", RowBox[List["j", " ", RowBox[List["(", RowBox[List["2", "+", "j"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "j"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["j", "!"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "24"], " ", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["3", " ", "n"]]]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "2"]]], "-", RowBox[List[FractionBox["1", "48"], " ", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "3"]]], "+", "\[Ellipsis]"]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "3"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02